Someone made a cheevos tool for iNaturalist
If you use iNaturalist and wish it was more like the Xbox 360, you can use this tool by a user named Glauber Ramos to track your progress across over 100 achievements they devised for the site.
To avoid doxing myself on iNat, I will show here a screenshot category example from a different user (specifically, the person on the site who has identified the largest number of spiders):
I'm sure that at this point, the reason that iNaturalist does not itself have cheevos would be because they'd considered the idea and decided not to pursue it. The site has been around too long for something like that to have not crossed their minds at some point.
However, I suspect that another good reason to not implement achievements on a website about observing nature would be to avoid motivating Bad Behavior In Nature. Glauber's achievement tool has achievements for observing endangered organisms, critically endangered organisms, and even stuff that is currently extinct in the wild (but presumably kept alive in a collection somewhere). I can't imagine that the site itself promoting this behavior would lead to a good place.
Glauber has a variety of other tools as well. Amusingly, the logo for the tool collection is a Pokedex, and there is a tool to take the top 100 species from any category in any location and create a "SpeciesDex" for that area to track your completion. This seems to be about as close as you could get to "the pokemon experience" for iNaturalist as an observer.
Cool stuff!!